Bifurcation for quasilinear elliptic singular BVP

Abstract

For a continuous function $g\geq 0$ on $(0,+\infty)$ (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in $\nabla u$, $-\Delta u +g(u)|\nabla u|^{2}$, with a power type nonlinearity, $\lambda u^{p}+ f_{0}(x)$. The range of values of the parameter $\lambda$ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of $g$ and on the exponent $p$. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.